Heat Transfer: A Practical Approach - Yunus A Cengel Fall 2003, Assignment 2 Friday, August 29, 2003 Chapter 2, Problem 62.

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1 Heat Tansfe: A Pactical Appoach - Yunus A Cengel Fall 003, Assignment Fiday, August 9, 003 Chapte, Poblem 6. Conside a steam pipe of length L = 5 ft, inne adius = in., oute adius =.4 in., and themal conductivity = 7. Btu/h ft F. Steam is flowing though the pipe at an aveage tempeatue of 50 F, and the aveage convection heat tansfe coefficient on the inne suface is given to be h =.5 Btu/h ft F. If the aveage tempeatue on the oute sufaces of the pipe is T = 60 F, (a) expess the diffeential equation and the bounday conditions fo steady one-- dimensional heat conduction though the pipe, (b) obtain a elation fo the vaiation of tempeatue in the pipe by solving the diffeential equation, and (c) evaluate the ate of heat loss fom the steam though the pipe. Answe: (c) 6,800 Btu/h Figue P-6 Chapte, Solution 6. A steam pipe is subjected to convection on the inne suface and to specified tempeatue on the oute suface. The mathematical fomulation, the vaiation of tempeatue in the pipe, and the ate of heat loss ae to be detemined fo steady one-dimensional heat tansfe. Assumptions Heat conduction is steady and one-dimensional since the pipe is long elative to its thicness, and thee is themal symmety about the cente line. Themal conductivity is constant. 3 Thee is no heat geneation in the pipe. Popeties The themal conductivity is given to be = 7. Btu/h ft F. Analysis (a) Noting that heat tansfe is one-dimensional in the adial diection, the mathematical fomulation of this poblem can be expessed as d d dt = 0 d and dt ( ) = ht [ T( d )] T ( ) = T = 60 F Steam 50 F h=.5 (b) Integating the diffeential equation once with espect to gives dt d = C T =60 F L = 5 ft Dividing both sides of the equation above by to bing it to a eadily integable fom and then integating, Copyight 003 The McGaw-Hill Companies Inc.

2 Heat Tansfe: A Pactical Appoach - Yunus A Cengel Fall 003, Assignment Fiday, August 9, 003 whee dt d C = T () = Cln+ C C and C ae abitay constants. Applying the bounday conditions give = : = + C ht [ ( C ln C )] = : T ( ) = C C = T Solving fo C C and C T T = h Substituting detemined to be C and C simultaneously gives and T T C = T Cln = T h ln into the geneal solution, the vaiation of tempeatue is T T T ( ) = C ln + T C ln = C(ln ln ) + T = T h (60 50) F = 60 F = F.4 7. Btu/h ft F.4 in.4 in (.5 Btu/h ft F)( / ft) (c) The ate of heat conduction though the pipe is dt C T T Q & = A = (πl) = πl d h (60 50) F = π (5 ft)(7. Btu/h ft F).4 7. Btu/h ft F (.5 Btu/h ft F)( / ft) = 6,800 Btu/h NOTE: if instead use h =.5 Btu/h-ft -F (as was stated in the poblem in text) then we get (60 50) F T ( ) = 60 F = F.4 7. Btu/h ft F.4 in.4 in (.5 Btu/h ft F)( / ft) and the heat tansfe ate is (60 50) F Q & = π (5 ft)(7. Btu/h ft F) = 758 Btu/h.4 7. Btu/h ft F (.5 Btu/h ft F)( / ft) Copyight 003 The McGaw-Hill Companies Inc.

3 Heat Tansfe: A Pactical Appoach - Yunus A Cengel Fall 003, Assignment Fiday, August 9, Chapte, Poblem 84. Conside a lage 5-cm-thic bass plate ( = W/m C) in which heat is geneated unifomly at a ate of 0 5 W/m 3. One side of the plate is insulated while the othe side is exposed to an envionment at 5 C with a heat tansfe coefficient of 44 W/m C. Explain whee in the plate the highest and the lowest tempeatues will occu, and detemine thei values. Figue P-84 Chapte, Solution 84. Heat is geneated unifomly in a lage bass plate. One side of the plate is insulated while the othe side is subjected to convection. The location and values of the highest and the lowest tempeatues in the plate ae to be detemined. Assumptions Heat tansfe is steady since thee is no indication of any change with time. Heat tansfe is one-dimensional since the plate is lage elative to its thicness, and thee is themal symmety about the cente plane 3 Themal conductivity is constant. 4 Heat geneation is unifom. Popeties The themal conductivity is given to be = W/m C. Analysis This insulated plate whose thicness is L is equivalent to one-half of an uninsulated plate whose thicness is L since the midplane of the uninsulated plate can be teated as insulated suface. The highest tempeatue will occu at the insulated suface while the Insulated lowest tempeatue will occu at the suface which is exposed to the envionment. Note that L in the following elations is the full thicness of the given plate since the insulated side epesents the cente suface of a plate whose thicness is doubled. The desied values ae detemined diectly fom T s T o 5 gl & ( 0 W/m )(0.05 m) = T + = 5 C + = 5.3 C h 44 W/m. C 5 3 gl & ( 0 W/m )(0.05 m) = Ts + = 5.3 C + = 54.5 C ( W/m. C) 3 g L=5 cm T =5 C h=44 W/m. C Copyight 003 The McGaw-Hill Companies Inc.

4 Heat Tansfe: A Pactical Appoach - Yunus A Cengel Fall 003, Assignment Fiday, August 9, Chapte, Poblem 89. Conside a homogeneous spheical piece of adioactive mateial of adius 0 = 0.04 m that is geneating heat at a constant ate of g = W/m 3. The heat geneated is dissipated to the envionment steadily. The oute suface of the sphee is maintained at a unifom tempeatue of 80 C and the themal conductivity of the sphee is = 5 W/m C. Assuming steady one-dimensional heat tansfe, (a) expess the diffeential equation and the bounday conditions fo heat conduction though the sphee, (b) obtain a elation fo the vaiation of tempeatue in the sphee by solving the diffeential equation, and ( c) detemine the tempeatue at the cente of the sphee. Chapte, Solution 89. Figue P-89 Heat is geneated unifomly in a spheical adioactive mateial with specified suface tempeatue. The mathematical fomulation, the vaiation of tempeatue in the sphee, and the cente tempeatue ae to be detemined fo steady one-dimensional heat tansfe. Assumptions Heat tansfe is steady since thee is no indication of any changes with time. Heat tansfe is one-dimensional since thee is themal symmety about the mid point. 3 Themal conductivity is constant. 4 Heat geneation is unifom. Popeties The themal conductivity is given to be = 5 W/m C. Analysis (a) Noting that heat tansfe is steady and one-dimensional in the adial diection, the mathematical fomulation of this poblem can be expessed as and d dt d d T ( 0 ) = = T s + = 0 with = constant 80 C (specified suface tempeatue) dt() 0 = 0 (themal symmety about the mid point) d ( b) Multiplying both sides of the diffeential equation by and eaanging gives d dt = d d T s =80 C g Integating with espect to gives 3 dt = + C ( a) d 3 Applying the bounday condition at the mid point, dt() 0 B.C. at = 0: 0 = 0+ C C = 0 d 3 0 o Copyight 003 The McGaw-Hill Companies Inc.

5 Heat Tansfe: A Pactical Appoach - Yunus A Cengel 5 Fall 003, Assignment Fiday, August 9, 003 Dividing both sides of Eq. (a) by to bing it to a eadily integable fom and integating, dt g d = & 3 g nd T () & a = + 6 C (b) Applying the othe bounday condition at = 0, B. C. at = 0 : T C C T s = 0 + = s Substituti ng this C elation into Eq. (b) and eaanging give T () = Ts + ( 0 ) 6 which is the desied solution fo the tempeatue distibution in the wie as a function of. (c) The tempeatue at the cente of the sphee ( = 0) is detemined by substituting the nown quantities to be (4 0 W/m )(0.04 m) T (0) = Ts + ( 0 0 ) = Ts + = 80 C + = 79 C (5 W/ m. C) Thus the tempeatue at cente will be about 7 C above the tempeatue of the oute suface of the sphee. Copyight 003 The McGaw-Hill Companies Inc.

6 Heat Tansfe: A Pactical Appoach - Yunus A Cengel Fall 003, Assignment Fiday, August 9, Chapte 3, Poblem 9. Conside a.-m-high and -m-wide double-pane window consisting of two 3-mm-thic layes of glass ( = 0.78 W/m C) sepaated by a -mm-wide stagnant ai space ( = 0.06 W/m C). Detemine the steady ate of heat tansfe though this doublepane window and the tempeatue of its inne suface fo a day duing which the oom is maintained at 4 C while the tempeatue of the outdoos is 5 C. Tae the convection heat tansfe coefficients on the inne and oute sufaces of the window to be h = 0 W/m C and h = 5 W/m C, and disegad any heat tansfe by adiation. Answes: 4 W, 9. C Figue 3.9 Chapte 3, Solution 9 A double-pane window consists of two 3-mm thic layes of glass sepaated by a -mm wide stagnant ai space. Fo specified indoos and outdoos tempeatues, the ate of heat loss though the window and the inne suface tempeatue of the window ae to be detemined. Assumptions Heat tansfe though the window is steady since the indoo and outdoo tempeatues emain constant at the specified values. Heat tansfe is one-dimensional since any significant tempeatue gadients will exist in the diection fom the indoos to the outdoos. 3 Themal conductivities of the glass and ai ae constant. 4 Heat tansfe by adiation is negligible. Popeties The themal conductivity of the glass and ai ae given to be glass = 0.78 W/m C and ai = 0.06 W/m C. Analysis The aea of the window and the individual esistances ae Ai R i o total A = (. m) ( m) = 4. m R = R R = R R R 3 = R = R = R ai, = = = C/W h A (0 W/m. C)(.4 m ) L m = Rglass = = = C/W A (0.78 W/m. C)(.4 m ) L 0.0 m = = = 0.93 C/W A (0.06 W/m. C)(.4 m ) o, = = = C/W o h A (5 W/m. C)(.4 m ) + R + R + R = (0.006) conv conv conv, = C/W conv, T R i R R R 3 R o T Copyight 003 The McGaw-Hill Companies Inc.

7 Heat Tansfe: A Pactical Appoach - Yunus A Cengel Fall 003, Assignment Fiday, August 9, 003 The steady ate of heat tansfe though window glass then becomes T T [4 ( 5)] C Q & = = = 4 W C/W R total The inne suface tempeatue of the window glass can be detemined fom T T Q & = T = T QR & conv, = 4 o C (4 W)(0.047 C/W) = 9. C R conv, 7 Copyight 003 The McGaw-Hill Companies Inc.

8 Heat Tansfe: A Pactical Appoach - Yunus A Cengel Fall 003, Assignment Fiday, August 9, Chapte 3, Poblem 4. Conside a powe tansisto that dissipates 0. W of powe in an envionment at 30 C. The tansisto is 0.4 cm long and has a diamete of 0.5 cm. Assuming heat to be tansfeed unifomly fom all sufaces, detemine (a) the amount of heat this esisto dissipates duing a 4-h peiod, in Wh; (b) the heat flux on the suface of the tansisto, in W/m ; and ( c) the suface tempeatue of the esisto fo a combined convection and adiation heat tansfe coefficient of 8 W/m C. Chapte 3, Solution 4 Figue P3-4 A powe tansisto dissipates 0. W of powe steadily in a specified envionment. The amount of heat dissipated in 4 h, the suface heat flux, and the suface tempeatue of the esisto ae to be detemined. Assumptions Steady opeating conditions exist. Heat is tansfeed unifomly fom all sufaces of the tansisto. Analysis (a) The amount of heat this tansisto dissipates duing a 4-hou peiod is Ai, Q = Q & t = (0. W)(4 h) = 4.8 Wh = Wh 30 C (b) The heat flux on the suface of the tansisto is πd A s = + πdl 4 π (0.005 m) = 4 Q& q& = = A s 0. W m + π (0.005 m)(0.004 m) = m = 959 W/m (c) The suface tempeatue of the tansisto can be detemined fom Q& (0. W) Q& = has Ts T ) Ts = T + = 30 C + ha (8 W/m - C)( m ( s = 38.8 C ) Powe Tansisto 0. W Copyight 003 The McGaw-Hill Companies Inc.